SEMICONDUCTOR DEVICE PHYSICS AND DESIGN

8.3. CURRENT-VOLTAGE CHARACTERISTICS 365

The current in the drain is given by (field =−dV /dx)

ID = channel area×(charge density)×(mobility)×(field)

= Z[h−W(x)]eNdμn

dV

dx

(8.3.1)

whereW(x)is the depletion width as shown in figure 8.8 andhis the channel thickness. Thus
h−W(x)is the channel opening. The depletion width at a pointxis given in terms of the gate
voltageVGS, the built-in voltageVbi, and the channel voltageV(x)by the depletion equation

W(x)=

[

2 [V(x)+Vbi−VGS]

eNd

] 1 / 2

(8.3.2)

To findIDas a function ofVDSandVGS, we substitute forW(x)in equation 8.3.1 and
integrate (IDis constant throughout the channel) to get

ID

∫L

0

dx=eμnNdZ

∫VDS

0

[

h−

{

2 [V(x)+Vbi−VGS]

eNd

} 1 / 2 ]

dV (8.3.3)

which gives (after dividing byL)

ID=

eμnNdZh

L

{

VDS−

2

[

(VDS+Vbi−VGS)^3 /^2 −(Vbi−VGS)^3 /^2

]

3(eNdh^2 / 2 )^1 /^2

}

(8.3.4)

We denote bygothe channel conductance when the channel is completely open,

go=

eμnNdZh

L

(8.3.5)

We have defined the pinch-off voltageVpas

Vp=

eNdh^2

2 

(8.3.6)

In terms ofVp, the drain current versus drain voltage characteristics can be written as

ID=go

{

VDS−

2

[

(VDS+Vbi−VGS)^3 /^2 −(Vbi−VGS)^3 /^2

]

3 Vp^1 /^2

}

(8.3.7)

Itmustberememberedthatthisequationwasderivedundertheconditionthatthegateanddrain
voltagesaresuchthatthereisnopinch-offnearthedrainregion,i.e.,

W(L)=

{

2 

VDS+Vbi−VGS

eNd

} 1 / 2

<h (8.3.8)